Abstract
We introduce and begin the study of new knot energies defined on knot diagrams. Physically, they model the internal energy of thin metallic solid tori squeezed between two parallel planes. Thus the knots considered can perform the second and third Reidemeister moves, but not the first one. The energy functionals considered are the sum of two terms, the uniformization term (which tends to make the curvature of the knot uniform) and the resistance term (which, in particular, forbids crossing changes). We define an infinite family of uniformization functionals, depending on an arbitrary smooth function f and study the simplest nontrivial case f(x) = x 2, obtaining neat normal forms (corresponding to minima of the functional) by making use of the Gauss representation of immersed curves, of the phase space of the pendulum, and of elliptic functions.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
S. Bryson, M. H. Freedman, Z.-X. He, and Z. Wang, “Möbius Invariance of Knot Energy,” Bull. Amer. Math. Soc. (N.S.) 28(1), 99–103 (1993).
M. H. Freedman and Z.-X. He, “Links of Tori and the Energy of Incompressible Flows,” Topology 30(2), 283–287 (1991).
M. H. Freedman, Z.-X. He, and Z. Wang, “Möbius Energy of Knots and Unknots,” Ann. of Math. 139(1), 1–50 (1994).
W. Fukuhara, Energy of a Knot, The fête of topology (Academic Press, 1988), pp. 443–451.
O. Karpenkov, “Energy of a Knot: Variational Principles,” Russ. J. Math. Phys. 9(3), 275–287 (2002).
O. Karpenkov, Energy of a Knot: Some New Aspects (Fundamental Mathematics Today, Nezavis. Mosk. Univ., Moscow 2003), pp. 214–223.
O. Karpenkov, “The Möbius Energy of Graphs,” Math. Notes 79(1–2), 134–138 (2006).
D. Kim and R. Kusner, “Torus Knots Extremizing the Möbius Energy,” Experiment. Math. 2(1), 1–9 (1993).
H. K. Moffat, “The Degree of Knottedness of Tangled Vortex Lines,” J. Fluid Mech. 35, 117–129 (1969).
J. O’Hara, “Energy of a knot,” Topology, 30(2), 241–247 (1991).
J. O’Hara, “Family of Energy Functionals of Knots,” Topology Appl. 48(2), 147–161 (1992).
J. O’Hara, “Energy Functionals of Knots II,” Topology Appl. 56(1), 45–61 (1994).
J. O’Hara, “Energy of Knots and Conformal Geometry,” K & E Series on Knots and Everything 33, World Scientific, 288 (2003).
A. B. Sossinsky, “Mechanical Normal Forms of Knots and Flat Knots,” Russ. J. Math. Phys. 18(2), (2011).
Author information
Authors and Affiliations
Corresponding author
Additional information
Oleg Karpenkov is partially supported by RFBR SS-709.2008.1 grant and by FWF grant no. M1273-N18. Alexey Sossinsky is partially supported by the RFBR-CNRS-a grant 10-01-93-111
Rights and permissions
About this article
Cite this article
Karpenkov, O., Sossinsky, A.B. Energies of knot diagrams. Russ. J. Math. Phys. 18, 306–317 (2011). https://doi.org/10.1134/S1061920811030046
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1061920811030046